Let's use logic to look at this. Cause it's fun.
The premises are: that logic is a system independent from premises, that logic is a system whereby we take premises and form new premises, that a premise may or may not be true.
If logic is a system independent from premises, and its output is premises, then we can say that the logical system does not influence the premises it outputs. Thus, a premise derived from a logical system and a set of premises only depends on which logical system and which premises are used. If this is so, a person who uses the same logical system and uses the same premises should logically derive the same output premise.
Now, logic is a system independent from premises. A premise being true or false is in itself a premise, and thus the derivation of a logical premise is not affected by the truthfulness of the starting premises. (The truthfulness of the output premise will be affected by the truthfulness of the starting premises, but not the logical system itself.)
Thus, a logical premise can be derived by anyone using the same logical system, regardless of whether or not they agree with the premises, so long as they actually know all the premises. If our brains are incapable of doing so, the premise will not be logical, and that can be demonstrated by independent evaluation, as a person using the same logical system will not be able to arrive at the same result.
Regarding the argument that arguing anything is pointless because of our inability to determine universal truth: Sure, except there are certainly levels of truth. There are things which are considered 'somewhat' universal - for example, the sky is blue, or Newton's Laws of Motion. Newton's Laws of Motion don't always apply, and they're not exact. But it's close enough to accurate to be readily and easily demonstratable. It's not universal truth, by any means, but it's a good, solid, workable model for terrestrial physics.
As we can't truly define a 'universal' truth, we can go from 'kinda universal' truths, and for the most part, they're exactly the same. Only problem is when you use 'kinda universal' truths and run into a situation where they don't apply. Things we consider 'true' now, like 'women and men should be given equal opprtunity' or 'independent expression is important', weren't always considered true, but that doesn't mean you can't use them as part of a logical argument, and that doesn't mean they're not a good basis to base a personal philosophy on. You just have to willing to adapt when new 'truths' come along.
Incidentally, this is why I fear the idea of eternal or very long life - humans aren't traditionally very good at discarding incorrect premises, and death has been the only way we've been able to discard premises that turn out to be false.
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